How to Teach Math — Abolish Math Courses!
A Sunday (July 29) New York Times opinion piece started me thinking about something I have often wondered about. Why are so many college students turned off on mathematics? The article's title was provocative: "Is Algebra Necessary?" The author, Andrew Hacker is an emeritus professor of political science at Queens College, City University of New York and author of the book, "How Colleges Are Wasting Our Money and Failing Our Kids — and What We Can Do About It". Well, that certainly got my attention.
Here is the gist. Our system of education forces all students (K-12) to take math courses and expects all students to complete some level of algebra. Algebra is considered the minimum level of math competency for college and we (society) have decided that essentially everyone is supposed to go to college. Even students in the humanities are expected to master some level of algebra. But here is the problem: Algebra turns most students off! Surprise, surprise.
Therein is the rub. Turn the students off because you are forcing them to learn something that purports to be a subject onto itself with the lame excuse that they will need it later (and, of course later never comes) or because it will help them think analytically/quantitatively (without explaining why that is important, or even what it is!) and you have lost them. I claim, unnecessarily.
Right now our country is in a panic. Fewer American students are choosing science and engineering as careers and they are doing more poorly on standardized math exams than kids from many other countries. We're in a panic because we hold firm to the idea that we have to compete with other countries economically. It is important, supposedly, to be number one when it comes to technology and science. And god forbid we fall behind in producing Nobel laureates (even if most of our prior laureates were actually foreign born!) Calls from politicians, business leaders, and social commentators (like Tom Friedman) extol us to knuckle down and do a better job in teaching kids math and science. The problem is that once we have turned kids off on math, we've lost them in science and engineering too. We are caught in a vicious cycle where we force feed more math only to make kids dislike it more and then we can only think to push it harder.
Hacker's article claims that kids don't need algebra. Indeed, they don't need most formal mathematics as taught in standalone courses. He argues, and I agree with him, that forcing kids to take such courses actually works to turn them off to school more generally and learning as a self-directed, internally motivated process. In short we are killing our education system by forcing kids to take really dry courses that are de-motivating and mentally numbing for the vast majority. There are certainly math-oriented individuals who like the topic as a topic and will go on to major in math as a subject onto itself. But these are rare persons. Most kids are not interested in memorizing the quadratic equation.
They are, however, intensely interested in relations! How does this relate to that? They are actually capable of thinking algebraically if given a chance to pursue that thinking in a motivating context.
What is Math Good For?
There have been a number of studies now that show that the vast majority of people, in their jobs, never actually use anything as sophisticated as algebra — explicitly. On the other hand almost everyone deals with relations and functions of variables to one degree or another. Anyone who has to convert proportions (e.g. in resizing a recipe) is faced with a simple form of algebraic relation.
Math is fundamental to measuring and transforming things. We all use it in one form or another, usually without realizing it. And all too often people face situations where they have to consider things like relative rates of change in processes that matter to them.In other words, math beyond arithmetic can be highly relevant. We just don't teach it that way.
We teach rules and axioms. We expect students to memorize facts that they will never explicitly use again in their lives. And in that process we turn them away from the very thing we dearly want them to learn. Talk about shooting one's self in the foot. It is true that math is foundational in the sense that our modern world operates on principles that can only be expressed meaningfully through mathematics. OK. So it is not unreasonable that children and young adults grasp the basics in order to function in this world and contribute to it. Even though calculators will do the arithmetic (I still think it is useful to teach arithmetic at an early age) the decisions about which operations to use, in what order, and on what variables, etc. is the kind of skill they need to develop. And it turns out many people do develop some rudimentary capabilities when their livelihoods depend on it.
Kids are born curious. They are born with the drive to find out why things are the way they are. They want to know how things relate to one another and what causes this and that to happen. They will naturally wonder “how much?” and “when?” and always “why?”. Our school system methodically beats that natural tendency out of them. By the time I see them in college they are programmed well to simply want to know what do they have to memorize to get a good grade in this course. Forget curiosity. Forget a drive to understand. The one thing they have been taught to understand is that in this system the GPA is the coin of the realm and their job is to maximize that quantity (see they are thinking mathematically even in spite of the system). I find myself agreeing with Hacker. We are killing the very thing we want to cultivate in the minds of our kids. We have to look at this whole thing with new eyes.
Many More Students Could be Scientists!
Many and perhaps most kids are natural born scientists. Anyone who has observed fourth graders on a field trip knows this is true. What happens then to kids as they grow older and get more schooling? The answer is that we are systematically squashing that natural curiosity and pounding any interest in understanding relations out of their minds with the way we go about schooling them.
There will always be students who by virtue of their superior intellects and indomitable curiosity will rise above the beating they get from schools. Such students will do well in math and science. They will go on to become engineers and scientists in spite of school, not because of it. But, unfortunately, there are so many more who are more fragile and are too easily intimidated by our overbearing force of subjects onto their minds. These kids are actually potentially capable of being scientist and engineers with something more nurturing than what our school system offers. They need to be allowed to mature at their own natural rate. And they need to see themselves as being able to consider problems from their own understanding rather than being told they need to memorize formulas. These are the vast majority. They do not need to be force fed mathematics to appreciate the natural world. They can learn mathematics in the context of their understanding that natural world.
And that is why I say abolish math courses! They do not really advance our objectives in having children and young adults understand the world. They intimidate kids. They stultify curiosity. Math is not a subject unto itself (unless you wish to become a theoretical mathematician!). It is a language for thinking about and expressing how the world works. It is really no different from spoken languages except that it is generally more precise. Once appreciated, the rules of relations in nature can become as natural to thinking as syntax is to natural languages.
The solution is to teach science better. By that I mean expose kids at an early age to the natural world with emphasis on how things relate to one another. Start exposing them to systems thinking early and reinforce it in every subject they take, even art and language classes. Start providing them with developmental stage-appropriate challenges, especially problems that can be studied and answered by a small group of peers. Let them discover the barriers to coming up with solutions that only some kind of math will get over. Let them puzzle on it for a while and then provide guidance with some examples of similar problems. If they are motivated because they really want to meet the challenge they will readily learn the math as a consequence of wanting to solve the bigger problem .
I became convinced of this approach while watching some seventh and eighth graders working with Lego robots to perform certain challenging tasks. These robots are an excellent exploratory environment for kids to try out ideas (hypothesize and experiment) and discovering how a little algebra can get the job done. The challenges involve multiple sensor inputs, light (with color values), touch, sonar readings, etc. The mobile robots have to perform some task such as following a black line on a white field while avoiding objects. This requires that their programs get varying values from the sensors and determine the significance of the values, especially relative to one another (for this age the programming environment is very friendly, a visual drag-and-drop construction tool kit). In other words they have to construct functions that produce motor output results in the proper fashion. Kids who wouldn't know an algebraic formula from their behinds are able to work out the relations and even come up with multiplicative factors that put variables in the proper scales. They naturally need some guidance to get started but they are soon working with these formulas and even trying variations and new formulas if they think they can improve a behavior.
The sciences, and I do mean all of the sciences not just the natural kind, are full of mathematical challenges that could provide the framework for learning math. Put the math where it belongs and let the kids have fun exploring the world they want desperately to know about. Stop teaching mathematics as a required subject in grade, middle, and high school. Embed math in the subjects they will take as the language they need to understand in order to communicate within that subject.
It is possible that by high school there will be some students who show a natural ability and interest in math as a subject. So there should be options in courses that resemble those now taught for these students to go further with formal mathematics. Quite possibly these are future math majors at the universities. There will always be a few who find pleasure in math just for its own sake and they should be given the opportunity to pursue that as well. They will be our future mathematicians and we need a few of them to work on pure math.
But the vast majority of students will only find interest in math if it is seen to serve a useful purpose for them. They need to experience math as part of a general problem solving process directed at challenges that truly interest them. This can extend to college as well. Students who take introductory sciences (e.g. Biology 101) should be exposed to more advanced algebra and even some calculus. They can be explicitly exposed to statistics when they do experiments, collect data, and then have to answer questions about the phenomenon they just witnessed. Ask a biology class to determine how much energy flows through a given ecosystem, first yearly, then monthly, then daily, then in minutes, etc. By doing this, exposing them to the notion of sample rates at smaller and smaller scales you've just started them down the road to understanding limits and infinitesimals. College level science courses (and many other subjects as well) could be designed to have a math learning skeleton upon which the flesh of the subject is hung. In other words the course designs would start with an outline of the mathematics that the community of educators agreed should be learned in the context of the subject. The professor's job becomes one of fashioning the subject content into challenges that are guaranteed to expose the need for those maths and then be ready to aid the students as they work through what techniques they need and how to use them. It is possible that an introductory science course might end up taking a whole year (2 semesters or 3 quarters) to complete because math is being taught within the context of the subject. So what? Students will learn both the math and the science much better as a result of learning them together in the context of solving complex challenging problems. It will take a different kind of professor than you typically see today (which is why this will never be adopted I suppose). The professor's job is to design problem-based curricula and act as a guide and mentor rather than a lecturing fount of all knowledge. I personally know a few professors who would be hard pressed to play that role. After all it is actually pretty easy to simply prepare lectures and grade assignments.
In our current paradigm we expect students to take the necessary math courses as prerequisites to taking the science courses. We expect them to have retained the math and be able to apply it as appropriate to the kinds of domain-specific problems the professor assigns. In other words, learn the math first, then take the science. We believe that is more efficient. Unfortunately you will hear many professors complain that the students are simply not prepared for the subject because they don't know the relevant math. It is easy to blame the high schools for failing to teach them what they need to know to be successful in the science courses. What no one seems able to see or understand is that the failure comes from the way we go about teaching math as a standalone subject. Even when a math text attempts to motivate a word problem with an example from “real life” it fails because as far as the student is concerned it isn't real. It is not the students. It is not the difficulty of the subject per se. It is the way we teach it that is the problem.
If we want to have more kids solidly learn math and science (and engineering) then we had better get our act together and stop beating them up with the subjects the way we are now. Of course it isn't just the math and sciences education that has to be radically reformed. The whole K-12 education system is based on efficiently moving kids through a system designed to produce worker bees. Then we all complain that kids can't think when they get out. For years reformers have been beating around the edges of true reform. They have been trying to fix things by tweaking the existing school system, curriculum, and pedagogy but never really taking a systems approach to designing a system that will truly educate. The proposal to abandon math courses and embed math within other subjects is a little more than a tweak. And as such it will buck the established system and thus be untenable. Moreover there is the problem of finding good science teachers who could actually pull this off. This is especially true at the K-12 level but, as I alluded to above, is also problematic at the college level. The kind of teaching that is called for is hard, highly skilled work. Perhaps if society recognized this they would agree to pay teachers more and thus attract those qualified people. Oh wait. I forgot, society is comprised of people who were schooled in the current paradigm! They are not likely to see the value in their children really learning because for the most part they never experienced it themselves. Another vicious cycle.
 It is true that learning math within a specific context has some problems with respect to transference to other situations where the same math would be applicable. We humans do have a tendency to not be able to naturally abstract or decontextualize instances of an exemplar. Kids could learn how to use a second-order polynomial function in one situation and completely fail to recognize how it might apply in a different setting. For example the exponential growth function is found applicable in so many different settings. If kids were to learn it in, say, a population biology setting, they might not recognize it use in an economics setting. But this is why I insist that systems science and thinking be taught everywhere. Once someone becomes more proficient in systems thinking it becomes much easier to use math abstractions in a variety of settings.
 And this extends to not just subjects that have science in their names, e.g. computer science, but to professional school subjects like business, social work, etc.